NRLF 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

>         ,, 

RECEIVED    BY    EXCHANGE 

\ 

Class 


£be  IHntversitE  of  Cbfcago 

FOUNDED  BY  JOHN  D.  ROCKEFELLER 


ON  THE  CONVERGENCY  OF  THE  SERIES  USED 
IN  THE  DETERMINATION  OF  THE  ELE- 
MENTS OF  PARABOLIC  ORBITS,  AND  THE 
ERRORS  INTRODUCED  IN  THE  ELEMENTS 
BY  IMPERFECTIONS  OF  THE  OBSERVA- 
TIONS 


A  DISSERTATION 

6 

SUBMITTED     TO    THE    FACULTIES    OF     THE    GRADUATE    SCHOOLS    OF    ARTS, 

LITERATURE,    AND    SCIENCE,    IN    CANDIDACY    FOR    THE 

DEGREE    OF    DOCTOR    OF    PHILOSOPHY 

(DEPARTMENT  OF  ASTRONOMY) 


BY 

WILLIAM  ALBERT  HAMILTON 


CHICAGO 

1903 


Gbe  "Gintvetsfts  of  Cbfcago 

FOUNDED  BY  JOHN  D.  ROCKEFELLER 


ON  THE  CONVERGENCY  OF  THE  SERIES  USED 
IN  THE  DETERMINATION  OF  THE  ELE- 
MENTS OF  PARABOLIC  ORBITS,  AND  THE 
ERRORS  INTRODUCED  IN  THE  ELEMENTS 
BY  IMPERFECTIONS  OF  THE  OBSERVA- 
TIONS 


A  DISSERTATION 

SUBMITTED     TO    THE    FACULTIES    OF     THE    GRADUATE    SCHOOLS    OF    ARTS, 

LITERATURE,    AND    SCIENCE,    IN    CANDIDACY    FOR    THE 

DEGREE    OF    DOCTOR    OF    PHILOSOPHY 

(DEPARTMENT  OF  ASTRONOMY) 


f  UNIVERSITY  1 

V  of  / 


BY 

WILLIAM  ALBERT  HAMILTON 


CHICAGO 
1903 


PRINTED  A  T  THE  UNIVERSITY  OF  CHIC  A  GO  PRESS 


PART  I. 

i.  Introductory. —  The  elements  of  the  orbit  of  a  comet  are  usually 
determined  by  means  of  the  data  obtained  at  three  separate,  complete 
observations ;  and  it  often  becomes  a  question  of  importance  to  the 
astronomer  as  to  the  suitability,  or  perhaps  we  might  say  the  suffi- 
ciency, of  such  a  set  of  observations  to  determine  with  accuracy  the 
required  elements.  He  is  confronted,  on  the  one  hand,  with  a  set  of 
formulae  somewhat  complex  in  their  nature,  and  which  are  subject  to 
limitations  in  their  application  owing  to  the  properties  of  the  various 
functions  involved  in  their  construction  ;  on  the  other  hand,  the  data 
of  observation  are  subject  to  limitations  owing  to  unavoidable  inac- 
curacies in  the  construction  of  the  telescope  and  the  multitude  of 
items  which  fall  under  the  class  known  as  errors.  It  is  thus  both  a 
mathematical  and  a  physical  problem  with  which  he  has  to  deal,  and  it 
becomes  important,  first,  that  a  careful  analysis  be  made  of  the  proper- 
ties of  the  formulas  and  the  conditions  under  which  they  may  be 
applied,  and,  secondly,  that  the  errors  which  present  themselves  in  the 
observations,  in  spite  of  the  greatest  care  and  skill  on  the  observer's 
part,  may  not  be  allowed  to  become  obscured  in  the  final  results  of  the 
computation.  It  has  been  the  purpose  of  the  study  of  which  this  paper 
is  a  partial  result  to  investigate  the  formulas  for  computing  cometary 
orbits  from  each  of  these  two  standpoints.  In  pursuance  of  this  plan, 
we  have  investigated,  among  other  questions,  the  nature  of  the  func- 
tions usually  known  as  the  " ratios  of  the  triangles;"  and  have  found 
the  precise  conditions  under  which  they  may  be  developed  into  con- 
verging power  series  of  the  time  intervals  between  the  observations.1 
This  discussion  is  given  in  the  first  part  of  this  paper.  In  another 
part  of  the  investigation  we  have  found  the  effects  of  the  errors  of  the 
observations  upon  the  computed  elements  of  the  orbit  of  a  comet, 
using  Obers's  method  as  a  basis  of  the  study.  To  this  is  added  the 
results  of  a  computation,  by  use  of  the  formulae  so  deduced,  of  the 
differentials  of  error  in  an  actual  case.  We  proceed  to  discuss  the  ratios 

i  That  the  series  converge  for  sufficiently  small  values  of  the  time -intervals  follows  from  CAUCHY'S 
existence  theorems  published  in  1842  (Coll.  Works,  ist  series,  Vol.  VII,  pp.  sf.).  PROFESSOR  PAUL 
HARZER  has  attained  the  same  general  results  in  the  Publications  of  the  Kiel  Observatory,  Vol.  XI, 
Part  2,  by  direct  treatment  of  the  series.  Evidently  the  results  of  both  Cauchy  and  Harzer  are  of  the 
nature  of  existence  theorems  and  were  not  intended  for  practical  use  by  computers,  because  the  true  radius 
of  convergence  was  not  found. 


183484 


4  CONVERGENCE  OF  SERIES  USED  IN  DETERMINATION 

of  the  triangles  and  convergency  of  the  series.     We  use  the  following 
notation  : 

2.  Notation. —  Let  /If  /2,  /3  denote  the  first,  second,  and  third  times 
of  observation,  respectively.  And  if  k  denote  the  Gaussian  constant 
and  m  the  mass  of  the  comet  in  terms  of  the  mass  of  the  sun  taken  as 
unity,  then  the  differential  equations  of  motion  of  the  comet  referred 
to  the  sun's  center  as  origin  of  co-ordinates  are, 


r3 


dt*  r3 

d*z  _   _6*(i  +  m)z 

where  r  is  the  heliocentric  distance  of  the  comet,  and  x,  y,  z  are  its 
rectangular  cartesian  co-ordinates.  In  all  practical  cases  m  will  be 
infinitesimal  in  comparison  with  the  mass  of  the  sun,  and  therefore  may 
be  neglected.  Furthermore,  if  we  so  change  the  unit  of  time  that  the 
new  unit  shall  be  equal  to  the  old  when  the  latter  has  been  multiplied 
by  k,  and  denote  the  time  when  expressed  in  the  new  units  by  /  —  /0, 
where  /0  is  any  particular  epoch,  we  may  express  these  equations  of 
motion  very  simply,  thus  : 


dt* 


In  these  equations  the  attractions  of  all  the  bodies  of  the  solar  system 
are  neglected,  except  that  of  the  sun. 

3.  Preliminary  notions.  —  Suppose  now  the  co-ordinates  of  the  comet 

at  the  time  /0  to  be  xot  y0,  zot  and  its  velocities  to  be—  =^,  -17,  -77  ; 

at      at     at 

then,  at  any  other  time,  the  co-ordinates  and  velocities  are  functions  of 
these  initial  conditions  and  /  —  4  ;  or,  as  we  say, 


,  0 

-j\x°>y<»  z°>  Tt'  dt*  dt'       °) 

with  similar  expressions  for  the  other  co-ordinates  and  the  velocities. 


f.,*v 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  5 

Now  it  is  known  from  the  theory  of  differential  equations  that  the 
co-ordinates  and  velocities  are  expansible  into  power  series  in  (/  —  f0)r 
of  the  form 


which  have  finite  radii  of  convergency,  if  r  does  not  vanish  for 
t  —  f0  =  o.1  In  the  partial  derivatives  above  t—  t0  is  to  be  placed 
equal  to  zero  after  differentiation.  Hence,  as  seen  from  (3), 


_8/J0~  dt   '  LBr\0~  dt" 

From  equation  (2)  we  obtain 

d*xQ  _    _  #o 

.'::';V  *"/;,'   ldx  « 

~dP"~ri~dt   ~Vl~dt    '  ' 

Equations  (5)  enable  us  to  find  the  coefficients  for  the  developments 
of  the  type  (3),  by  means  of  which  the  co-ordinates  and  velocities  of 
the  comet  at  any  time  /  are  expressed  as  power  series  of  the  time  inter- 
vals /  —  /0,  and  coefficients  depending  only  upon  the  co-ordinates  and 
velocities  at  the  initial  time  /0.  By  means  of  these  developments  of 
the  co-ordinates  the  so-called  ratios  of  the  triangles  are  built  up  in  the 
form  of  series  which  depend  upon  particular  time  intervals  selected 
from  those  determined  by  the  three  observations.  It  is  in  regard  to 
these  latter  series  that  we  wish  to  find  the  conditions  of  convergency; 
and  it  is  at  once  evident  that  their  convergency  will  depend  upon  the 
convergency  of  the  series  of  the  type  (3),  since  the  ratios  of  the  trian- 
gles are  functions  of  the  co-ordinates  alone. 

4.  Convergency  of  series. —  From  well-known  theorems  of  the  theory 
of  functions  of  complex  variables  it  is  clear  that  any  expansions  what- 
ever of  the  ratios  of  the  triangles  into  power  series  for  given  time 
intervals  and  initial  conditions  cannot  have  greater  radii  of  conver- 
gency than  the  values  which  are  determined  by  the  poles  and  branch 
points  of  the  expressions  of  those  ratios  as  functions  of  the  time  inter- 
vals. First,  however,  we  study  the  nature  of  the  functions  which 
express  x,  y,  z  in  terms  of  /,  and  from  these  find  the  true  radius  of 
convergency. 

1  JORDAN,  Cours  d"1  analyse,  Vol.  Ill,  p.  99. 


6  CONVERGENCE  OF  SERIES  USED  IN  DETERMINATION 

5.   Co-ordinates  as  functions  of  the  time.  —  From   the  geometrical 
relations  of  the  orbit  of  the  comet  we  have  the  relations 

x=  r  [cos  (v  -f-  o>)  cos  li  —  sin  (v  +  w)  sin  li  cos  i]  , 

y  =  r  [cos  (v  -j-  o>)  sin  O  -j-  sin  (v  -f-  <o)  cos  O  cos  /']  ,  (  6  ) 

z  =  r  [sin  (v  -f-  w)  sin  /]  , 

where  v  is  the  true  anomaly,  o>  is  the  argument  of  latitude  of  the  peri- 
helion, O  is  the  longitude  of  the  node,  and  /  is  the  inclination  of  the 
orbit  to  the  ecliptic.  The  last  three  quantities  are  independent  of  the 
time  ;  while  v  and  r  are  expressible  in  terms  of  /  by  means  of  the  rela- 
tions 


—- 

l  +  COS  V 


where/  is  the  latus  rectum  of  the  parabolic  orbit  of  the  comet  and  II 
is  the  time  of  perihelion  passage.  II  and  /  are  thought  of  as  expressed 
in  units  described  in  section  2  above  —  a  usage  which  we  shall  con- 
tinue throughout  this  paper. 

6.  The  solution  of  the  cubic.  —  By  means  of  the  equation  (7)  we  are 
enabled  to  express  x,  y,  and  z  in  terms  of  the  time  intervals  /  —  II.  In 
order  to  do  this  we  introduce  the  auxiliaries 


<£  —  tan-  . 

2 

Then  the  last  equation  of  (7)  becomes 

<£3  +  3<£  —  2T  =  o  .  (9) 

This  is  the  so-called  normal  form  of  the  cubic  in  the  quantity  <£.     Its 
solutions  by  Cardan's  formula  are 


(10) 


where  ^z—  (r-J-  j/i-f-r2)*,  q2  =  (T  —  i/i  +  T2)*,  and  i,  e,  «2  are  cube 
roots  of  unity.1 

i  See  BURNSIDE  AND  PANTON,  Theory  of  Equations,  p.  108. 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  1 

7.  Branch  points.  —  We  wish  to  express  <f>  in  a  power  series  in  /,  and 
must,  therefore,  find  the  branch  points  and  poles  of  the  function.     At 
once  we  have  the  branch  points  r  =  i  and  T=  —  i,  where  i=i/  —  i. 
Also  T  =  oo  is  a  branch  point,  as  is  easily  seen  by  putting  r  =  —,  ,  and 

letting  T'  approach  zero.  This  is  the  same  as  putting  r=  oo  ,  and  we 
easily  find  that  all  three  solutions  have  the  same  value  at  this  point. 
If  now  we  consider  a  Riemann  surface  of  three  sheets  with  branchpoints 
at  T  =  /,  T  =  —  /,  T  —  oo  ,  then  by  the  theory  of  functions  of  a  complex 
variable  we  know  that  the  quantity  <j>  is  a  uniform  function  of  position 
on  this  surface. 

8.  Connection  of  the  sheets.  —  In  order  to  get  a  clear  idea  of  the  sur- 
face, it  is  necessary  to  find  what  sheets  pass  into  each  other  at  the  two 
branch  points  which  are  in  the  finite  part  of  the  plane.    To  do  this  we 
need  to  follow  only  the  purely  imaginary  values  of  T;  for  the  two 
branch  points  in  question  are  on  the  axis  of  pure  imaginaries.    Indeed, 
we  may  also  consider  the  branch  point  T  =  00  to  be  on  this  same  axis. 

In  order  to  simplify  matters  and  at  the  same  time  render  the  rea- 
soning clearer  we  make  the  transformation 

T  =  /  cos  0  ,  (u) 

where  6  is  real  or  complex.     Then  q^  and  q2  become 

_z* 

q^  =  [/(cos  6  —  i  sin  0)]*  =  —  ie    3  , 

16 

qz  —  [/(cos  0  -f  i  sin  0)]i  =  —  ie*  . 

yirz  _  ziti 

And  since  we  may  write  e  —  e  3  ,  **=  e     s  ,  we  obtain  from  (10) 
/«       _«\  ! 

<k  —  —  /  \/3  +  e      3J  =  —  2t  COS  3  , 

.  /!•(*-„)          -!'(*-„  A  ^ 

<k>—  —  t\&  +  <?     3  /  =  —  21  COS     3      ,  (12) 


<#>3  =  —  /  \<?3  +  e   3          y  =  _  2*  cos  3    . 

Now  from  (n),  if  0  takes  real  values,  T  is  purely  imaginary  and 
takes  values  between  T  =  I  and  T—  —  /;  while  if  6  is  a  pure  imaginary, 
T  takes  pure  imaginary  values  with  moduli  greater  than  unity.  Only 
when  0  is  complex  does  T  take  real  or  complex  values.  Hence  for  our 
purpose  we  need  consider  only  imaginary  values  of  0,  or  real  values  of 
0,  in  order  to  find  the  connection  of  the  sheets. 


8  CONVERGENCE  OF  SERIES  USED  IN  DETERMINATION 

We  must  notice  also  that  T  is  a  periodic  function  of  0 ;  hence  when 
we  wish  T  to  trace  the  line  between  the  two  branch  points  but  once,  we 
take  the  primitive  period  and  consider  this  alone.  Now,  in  order  that 
T  may  take  only  pure  imaginary  values  while  passing  from  T  =  —  / 
to  T  —  /,  6  must  take  the  real  values  between  o  and  TT,  and  therefore 

-  will  take  the  real  values  between  o  and  - .     We  get  the  following 
3  «j 

correspondence  for  0,  T,  <k,  <£2,  and  <£3: 

6  T  0!  0g  03 

o  i        —  21        i  i 

o       — 

7T  —  /  —I  —I  21 

Denote  the  branch  points  T  =  /  and  T=  —  /  by  ^4  and  ^4;  respec- 
tively. Then  from  the  table  above  we  find,  according  to  the  period 
selected,  that  the  two  values  <f>2  and  <£3  become  equal  when  T  approaches 
A;  but  when  T  arrives  at  /4'  along  the  path  selected,  this  does  not 
repeat  itself;  but  instead  we  have  <k  ==<&,.  Hence  sheets  <£2  and  <f>3 
are  connected  at  r  =  i;  while  <f>2  and  <k  are  connected  at  T=  —  /.  It 
follows  that  if  we  start  at  r  =  o  in  the  r-surface  and  make  a  complete 
circuit  once  around  A,  then  o  and  /  1/3  will  change  places;  while  for 
a  circuit  around  A',  o  and  —  iV 3  will  change  places.  If  we  draw 
branch  cuts  from  A  to  infinity  and  from  A '  to  negative  infinity,  the 
continuation  of  the  sheets  when  crossing  these  cuts  will  be : 

Along  A    to       oo 


Along  A '  to  —  oo  , 


1.3-2 
1-2-3 


All  three  sheets  are  connected  at  T  =  oo  .     A  section  along  the  axis  of 
pure  imaginaries  will  appear  as  in  Fig.  i. 


r=i 

FIG.  i. 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  9 

It  will  be  of  importance  for  what  follows  to  notice  that  in  the 
0-plane  the  portion  which  is  bounded  by  the  axis  of  pure  imaginaries 
and  the  line  0  =  ir  is  a  conform  representation  of  the  whole  r-plane, 
each  sheet  being  represented  once  in  the  fundamental  region. 

9.  Poles  of  <£  in  the  sheets. — It  is  well  known  that  where  z  is  a  com- 
plex variable,  the  function  f  can  become  infinite  only  for  infinite 
values  of  z.     Hence  it  follows  from  (12)  that  <j>  cannot  become  infinite 
except  for  infinite  values  of  0.     Moreover,  owing  to  the  periodicity  of 
the  function  <?z,  which  makes  ezJr™*  =  ez,  the  above  infinite  value  of  9 
must  be  either  purely  imaginary  or  perhaps  complex  with  the  imagi- 
nary part  of  the  complex  expression  infinitely  great.     But,  from  (u), 
such  a  value  of  0  gives  T  infinite.     Hence  it  follows  that  <f>  cannot 
become  infinite  except  for  infinite  values  of  T.    Hence  there  are  no 
finite  poles  of  <£  in  the  sheets  of  the  Riemann  surface. 

10.  Zeroes  in  the  sheets. —  By  use  of  (12)  we  are  also  enabled  to  find 
at  once  the  zeroes  of  <#>  in  the  r-surface.     The  general  condition  for  the 
vanishing  of  <j>  is  given  by  either  of  the  conditions 

e 

cos  -  =  o  , 


cos 


These  are  virtually  the  same,  since  we  may  get  the  one  from  the  other 
by  putting  0=  0±  2*.     It  is  then  only  necessary  to  find  the  value  of  6 

e 

for  which  cos-=o.     Now,  by  methods  well  known  to  the  theory  of 

trigonometry1  it  is  readily  proved  that  the  only  values  of  6,  real  or  com- 
plex, which  satisfy  this  condition  are 


where  n  is  a  positive  or  negative  integer  or  zero.  It  follows  that  only 
one  zero  of  <£  is  to  be  found  in  each  fundamental  region  of  the  0-plane 
for  each  of  the  sheets  on  the  Riemann  surface.  Thus,  for  the  first 

sheet,  it  is  that  value  of  r  which  corresponds  to  the  value  of  6  =  - 

which  gives  by  use  of  (n)  T  —  o  .  We  have  already  seen  in  the  table 
following  (12)  that  only  one  branch  of  <£  vanishes  at  this  point ;  and 

i  See  CHRYSTAL,  Algebra,  Vol.  II,  chap.  29. 


10  CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 

that  the  particular  one  which  vanishes  thus  is  dependent  entirely  upon 
the  sheet  of  the  Riemann  surface  in  which  T  is  found. 

ii.  Resume. —  We  note  here  the  following  summary  of  results  as  to 
critical  points  upon  the  Riemann  surface  upon  which  <£  is  a  function  of 
position  : 

(  T  =  o  in  the  r-surface, 

Zeroes  at  •<  ,,       *••>,*  r^/ii 

)  6  =  -  in  the  fundamental  region  of  the  0-plane  ; 

»       I  2 

Poles,  none  in  the  finite  part  of  the  r-plane;  (13) 


Branch  points  <  T  =  —  /  , 

[    T=°°      . 

1 2.  Rational  functions  of  <£  «#</  T. —  At  this  place  we  state  the  follow- 
ing theorem  which  will  be  useful  for  later  work : 

Every  rational  function  of  $  and  T  is  a  uniform  function  of  position 
on  the  same  Riemann  surface  as  that  which  describes  <f>  as  a  function  of 
T  and  its  branch  points  are  at  the  same  places?  It  is  to  be  remembered, 
however,  that  this  theorem  does  not  apply  to  the  zeroes  and  poles  of 
such  a  rational  function  of  <£  and  T.  These  may  be  located  otherwise 
than  as  described  in  (13). 

13.  Co-ordinates  x  and  y  as  functions  of  r. —  We  may  write  the  first 
two  equations  of  (6)  as  follows: 

V  27 

T/  v\ 

y  =  [£  cos  v  —  jz  sin  v]  1 1  -}-  tan*  -1   ; 

where/  =  r(i  +  cos  v)  ;  and 

2C 

—  =  cos  o>  cos  II  —  sin  w  sin  11  cos  i  , 
P 

2S 

—  =  sin  <D  cos  O  -f-  cos  c>  sin  H  cos  *  , 
P 

2C 

—  =  cos  CD  sin  O  +  sin  <o  cos  O  cos  /  , 
P 

2  c 

— -  =  sin  CD  sin  11  —  cos  <o  cos  11  cos  i  . 
P 

i  See  FORSYTHE,  Theory  of  Functions,  p.  369. 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  1  1 

v  v 

i  —  tan2  -  2  tan  - 

Using  the  relations  cos  v  =  -  —  and  sin  v  —  -    -  ,  we  may 

i  -f  tanr  -  i  -f-  tan*  - 

2  2 

write  (14),  where  we  put  tan-  =  <£  in  the  form, 

X  =  C  —  2  S<f>—  C<p 
,=  ,,-2,,*-,,^' 

Now,  in  the  equations  (16)  c,  clt  s,  and  sx  are  constants  independent  of 
T,  and,  by  the  theorem  given  in  the  last  article,  x  and  y,  considered  as 
functions  of  T,  are  functions  of  position  on  the  same  Riemann  surface 
which  defines  <f>  as  a  function  of  T,  and  the  branch  points  of  x  and  y  are 
r  =  ij  r  =  —  i,  and  r=x>  .  Moreover,  since  c,  c^  s,  and  st  are  constants 
and  never  infinite,  x  and  y  cannot  become  infinite  except  where  <£ 
becomes  infinite,  viz.,  at  T=  oo  .  Hence  we  have  : 

Theorem:  x  and  y  have  poles  in  the  Riemann  sheets  only  at  T—  w 
and  they  have  branch  points  at  r  =  i,  T==—  /,  and  T  =  <X>. 

It  follows  from  the  above  that  x  andj>  are  holomorphic  functions  of 
T  in  the  sheets  of  the  Riemann  surface  except  at  points  T  =  I,  r=  —  i, 
and  T  =  oo  .  Therefore  they  may  each  be  expanded  into  power  series 
with  argument  T  —  r0  in  the  vicinity  of  any  point  T  =  TO  .  These  series 
will  be  convergent  inside  of  a  circle  whose  center  is  at  TO  and  whose 
radius  reaches  from  TO  to  the  nearest  of  the  points  T  =  /,  or  T  =  —  /. 

14.  Radius  of  convergency.  —  If  in  (3)  we  replace  /  and  /0  by  their 
corresponding  values  in  T  by  relation  (8),  we  have  just  such  an  expan- 
sion as  described  in  the  last  article.  If  we  should  at  the  same  time 
take  /0  —  o,  the  expansion  in  x  becomes  of  the  form 


where  a0,  0X  ,  etc.,  are  constants.  This  series  will  be  convergent  inside 
a  circle  whose  center  is  T  =  o  ,  and  with  radius  unity  reaching  up  to 
the  branch  points  T  =  i  and  T  —  —  i.  Hence,  the  true  radius  of  con- 
vergency in  this  case  would  be  |r|=  i;  or  from  (8) 

c—  )=*  •  <"> 

•3 


Suppose  in  (17)  we  give  to  /  any  value,  say  i,  which  would  corre- 
spond to  a  perihelion  distance  of  0.5  ;  then  if  we  make  £==  —  ,  which 
is  approximately  its  value,  the  case  under  supposition  would  give  as  the 


12  CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 

limit  for  the  time  interval  for  which  the  expansion  of  x  into  power 
series  would  be  convergent,  the  value  20  days.  The  same  period  would 
hold  for  the  corresponding  expansion  of  y. 

If  TO  were  any  finite  point  not  equal  to  zero,  say  some  point  on  the 
real  axis  of  the  r-plane,  then  the  radius  of  the  true  circle  of  conver- 
gency  would  be  larger  than  that  given  above.  In  this  case  the  radius 
would  be  1?T=  V  i  -f-  T*,  which  holds  for  both  the  x  and  the  y  series. 
The  radius  of  convergency  of  the  corresponding  series  in  /  is  at  once 
deducible  from  the  series  in  r  through  the  relation  (8).  The  relation 
is  always 

•Rt=   —   XT    , 

where  the  subscripts  denote  the  argument  of  the  series.  Since  in  an 
absolutely  convergent  series  we  are  at  liberty  to  change  the  order  of 
the  terms  at  will,  we  may  express  (3)  and  the  corresponding  equation 
my  by  use  of  coefficients  of  the  kind  given  in  (5),  as  follows  : 

dx 


where  A  and  B  for  their  first  few  terms  are 

A-    i?i'+i--°+-r-  i^Y+i^ 

2rl^2ridt  "h24U       r\\dt  )  ^  ri  dt' 
B      /      "3  +  "4^4 

=  /  -673  +  47;*- 

In  these  series  we  have  taken  /0  =  o.  They  may  be  written  as  series  in 
t  —  t0  by  use  of  the  theory  of  continuation  of  power  series. 

Since  a  power  series  serves  in  every  way  to  define  the  behavior  of 
the  function  from  which  it  is  derived  so  long  as  we  remain  within  its 
circle  of  convergence,  we  can  deal  with  the  series  (19)  as  with  quanti- 
ties which  obey  all  the  laws  of  ordinary  algebra  —  association,  commu- 
tation, etc.;  such  as  ordinary  polynomials  or  rational  quantities,  and 
the  resulting  series  will  be  convergent.1 

15.  Ratios  of  the  triangles.  —  We  denote  the  triangle  between  the 
positions  of  two  radii  vectores  of  the  comet's  orbit  by  the  expression 
[>i,  ^J,  where  /  and  j  denote  the  order  of  any  two  of  the  three  obser- 
vations ;  also  in  general  we  denote  the  co-ordinates  of  the  first,  second, 

*  See  CHRVSTAL,  Algebra,  Vol.  II,  pp.  139-43. 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  1 3 

and  third  observations  by  the  subscripts  i,  2,  3  respectively.  Now  the 
ratios  of  the  triangles  [rit  r^\  are  equal  to  the  ratios  of  their  projections 
upon  any  plane,  which  fact  may  be  expressed  thus : 


Let  now  xatya,  -r^-r  De  taken  as  the  zero  values  of  the  co-ordinates 
at    at 

and  velocities  in  the  expansions  (19)  and  (20).     Then  we  get 


i     \ 
dx  (22) 


where  At  ,  B^,  A3,  B^  are  defined  by  : 


Now,  the  series  (22)  are  convergent   within    the  same  circle.     It 
follows  that,  since  xa,yat  -j-2,  -^  are  not  in  general  equal  to  zero,  the 

series  (23)  are  also  convergent  in  this  same  circle.1  It  follows  that 
since  for  such  series  the  law  of  distribution  holds2  the  products  Alt  B3 
and  A3  B^  are  also  convergent  series.  Hence,  from  the  law  of  addition, 
we  have  that  A^B3  —  B*A3  is  also  convergent.3 

We  are  now  at  liberty  to  substitute  the  values  of  xs,  ylt  x3,y3  as 
given  by  (22)  in  the  ratios  on  the  right  of  (21).     We  get,  after  the 

substitution  indicated  and  by  canceling  the  factor  *«rs  ~-^~v7 
the  two  members  of  the  ratio, 

i  Ibid.,  p.  178,  5.  2  ibid.,  pp.  142,  143.  3  Ibid,  p.  141. 


14          CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 

[>..  'J=    -g3=(/s~^)rI    i(/3-/2)2-(A-/2)2 . 

[rf,  r.]  ^       (/x-/a)L        6  ,* 

4"      "^    "i2'"]'  ,   } 

[r»rj=          -^         =(A-/.)r     ,    i  (/3 -/.)"-(/.-/.)• 


4  ^  ^/ 

where  ^r,   ^x,  ^43,  jff3  have  the  meaning  given  by  (23). 

Now,  B^  and  ^3  are  series  with  arguments  /,  —  /2 ,  and  /3  —  /2 
respectively.  They  hold,  /.  ^.,  are  convergent,  as  long  as  /x—  /2  and 
/3—  /2  obey  the  relations 


(25) 


Also  the  series  A1^3—  B^A^  which  has  two  arguments,  viz.,  /t—  /2 
and  /3  —  /2  is  convergent  so  long  as  (25)  holds. 

16.  The  zeroes  of  B^  and  A^B^—  B^A3.  —  If  Bt  should  vanish,  or  if 
A^B3—  BtA3  should  vanish,  then  the  fractions  on  the  right  of  (24) 
evidently  would  no  longer  be  legitimate.  It  is  easily  seen  that  the 
former  contingency  cannot  occur  unless  /r—  /2=o.  As  to  the  latter, 
we  state  two  theorems  which  are  readily  proved,  but  the  proofs  of 
which  we  will  here  omit.  They  are  stated  as  follows  : 

Theorem  I:  The  expression  xTy3—  ytx3  can  vanish  only  for  the 
real  values  of  v3—  v,. 

Theorem  II:  In  all  cases  where  the  times  of  the  observations  are 
distinct  and  where  the  difference  of  the  longitudes  of  the  comet  in 
its  orbit  is  not  equal  to  an  odd  multiple  of  TT,  the  expression  (f,,o) 

n  7? 

cannot  vanish  and  the  expressions  -W,-T-B  —  '  p       are  legitimate  frac- 

z>x   A1±f3  —  £>tA3 

tions  which  may  be  expressed  as  series,  each  of  which  is  convergent  for 
all  cases  where  |/3—  /2|<—  T/r2  -f-  i  and  |  A  —  /2  1  <  —  I/  T,  -f  i  . 

O  .7 

The  first  terms  of  these  series  are  written  out  in  the  right  members  of 
(24).  In  many  cases,  however,  the  computer  may  prefer  to  use  the 
fractions,  and  these  are  always  the  safer  formulae  when  doubt  in  any 
way  exists  as  to  their  legitimacy. 


OF  ELEMENTS  OF  PARABOLIC  ORBITS 


17.   Computation  by  use  of  the  series. —  The  fractions  on  the   right 
of   (24)   have    both    numerators    and    denominators   in  the  form    of 

p\    / 

series.     Their    radius   of    convergency    is  Rt  =  —  V  i  -f-  ^  ,     where 

O 

r2  =  -4  (/2  —  II).     If  we  make  &=  —  ,  the  following  table  will  give  cor- 
pt  oo 

responding  maximum  intervals  of  time  for  different  values  of  /  for 
which  the  series  are  convergent  when  t2  is  taken  equal  to  II : 

TABLE  I. 


4 

2 

160.0 


103.4 


2-5 

i.2S 
79-8 


2. 
56.0 


0.75 
36.2 


1-25 
0.62 
27.4 


i. 
o.S 

20.0  14 


0.25 
0.12 
2-5 


1.8 


0.6 


0-44 


.08  o 

0.05  0.04  0.02  0.01 


0.22  0.06 


It  is  evident  that  for  any  particular  value  of  p  the  time  intervals 
should  be  well  within  the  limit  of  values  for  which  the  series  are  con- 
vergent. This  is  especially  true  if  we  would  have  the  most  rapid  con- 
vergence—  a  thing  most  desirable  from  the  standpoint  of  the  computer. 
In  fact,  as  is  well  known,  it  is  imperative  to  have  this  convergence  so 
rapid  that  at  most  but  one  or  two  terms  will  give  sufficiently  approxi- 
mate values  of  the  ratios.  The  reason  for  this  is  at  once  evident 
when  we  consider  that  the  series  are  transcendental  in  character. 

Thus,  the  quantities  — ^  and  r2  which  enter  into  the  terms  higher  than 

the  first  are  essentially  unknown  from  the  start,  and  cannot  even  be 
guessed  at  with  any  degree  of  certainty  until  an  approximate  value  of 
p  has  been  obtained.  It  cannot  be  too  strongly  insisted  upon,  there- 
fore, that,  in  order  to  get  the  closest  determination  of  the  ratios  of  the 
triangles,  the  greatest  care  must  be  taken  to  secure  a  set  of  time  inter- 
vals which,  by  their  co-ordination  with  the  parameter  of  the  orbit  in 
hand,  will  make  the  series  rapidly  convergent.  It  is  true  that  this  is 
more  or  less  a  question  of  trial  to  start  with ;  yet,  when  a  value  of  p 
has  been  once  computed  by  means  of  any  set  of  time  intervals,  it  will 
be  seen  at  once  whether  the  value  so  obtained  is  one  for  which  the 
series  are  sufficiently  convergent  for  the  time  intervals  employed.  If 
this  is  not  the  case,  then  new  time  intervals  should  be  taken  and  the 

computation  made  over  again. 

. 

- 


PART  II. 

IN  this  part  of  this  paper  are  deduced  differential  equations  of 
relation  which  give  the  errors  of  the  computed  elements  of  the  comet- 
ary  orbit  expressed  in  terms  of  the  errors  of  the  observations.  At  the 
close  is  also  given  the  results  of  a  computation  in  which  the  formulae 
are  applied  to  an  actual  example. 

1 8.  Geometrical  relations.  —  Let  p  represent  the  geocentric  distance 
of  the  comet,  A  and  ft  its  geocentric  longitude  and  latitude,  respec- 
tively. Let  R  represent  the  heliocentric  distance  of  the  earth,  Z  and 
B  the  longitude  and  latitude  of  the  sun,  respectively.  The  subscripts 
i,  2,  3  are  used  as  before  to  represent  the  order  of  the  observation  to 
which  the  corresponding  co-ordinate  applies.  With  this  notation  any 
one  of  the  following  three  equations  will  express  the  relation  between 
the  geocentric  distances  of  the  comet  at  the  first  and  third  observations. 
For  their  derivation  the  reader  is  referred  to  Moulton's  Introduction  to 
Celestial  Mechanics,  chap.  x. 

p3  =  mf  +  M'Pl^  (i) 

where  m '  and  M'  are  defined  by 


rt-iiJz' 

^-zlf/^rj  ' 


cos  ft  sin  (\3—L2)  —  sin  ft  cos  ft  sin  (A2 
,      [V, ,  rj  sin  ft  cos  ft  sin  (A2  —  Z2)  —  cos  ft  sin  (AT  —  ZT)  (10) 

-  fc ,  rj  cos  ft  sin  (A3  —  Z2)  —  sin  ft  cos  ft  sin  (A2  —  Z2)   ' 
Z/  =  £z  sin  (ZT  —  Z2),     L3=£3  =  sin  (Z3  —  Z3)  . 

where  w"  and  M"  are  defined  by 

„      sin  ft  [|>2 ,  rj  ^  cos  (Zt  -  Z2)  +  [rt ,  r^  R3  cos  (Z, — Z2)  -  [V, ,  r  J  /?2]  ^ 
[rx ,  rj  [sin  ft  cos  ft  cos  (A3  —  Z2)  —  sin  ft  cos  &  cos  (A2  —  Z2)] 

jlf  —  :  2)    3:  sin  ft  cos  ft  cos  (A3  —  Z2)  —  sin  ft  cos  ft  cos  (\0  -Z2)  . 

^*I  >     ^2_ 

where  wr//  and  M'"  are  defined  by 
sin  (A2  -  A,)  cos  ft 


_ 


,,  rJsin(A3-A2)cos 


t,  rJcosftsin(A3-A2) 
16 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  1  7 

19.  Generalities.  —  Equations  (i),  (2),  and  (3)  involve  the  dynami- 
cal law  that  the  motion  of  a  comet  is  in  a  plane  passing  through  the 
center  of  the  sun.  The  quantities  m'  ',  mn  ',  m'"  ,  M  ',  M'  ',  J/'"  are 
functions  of  the  geocentric  longitudes  and  latitudes  A,,  ft,  A2,  etc., 
and  of  the  ratios  of  the  triangles.  They  also  involve  R*  ,  Rz,  R^  and 
the  longitude  and  latitudes  of  the  sun.  But  /?,,/?,,  Z,,  ^2,  etc.,  are 
independent  of  the  errors  of  observation,  since  they  are  taken  from  the 
Ephemeris.  Likewise  the  time  intervals  which  are  involved  in  the 
ratios  are  independent  of  A,,  ft,  A2,  ft,  A3,  ft,  which  are  derived 
directly  from  the  right  ascension  and  declination  determined  by  the 
settings  of  the  instrument.  Owing  to  this  last  fact,  we  shall  speak  of 
A-D  A.2,  ft,  etc.,  as  observed  co-ordinates. 

In  practice  some  one  of  the  three  equations  (i),  (2),  (3)  is  usually 
more  advantageous  than  the  others  owing  to  the  particular  problem  of 
computation  at  hand.  For  the  method  of  determining  the  one  to  be 
used,  the  reader  is  again  referred  to  the  same  treatise  and  chapter  as 
in  the  last  article.  For  our  purpose  here,  the  fact  just  stated  is  of 
interest  because  it  necessitates  the  derivation  of  a  separate  set  of 
formulae  for  the  errors  in  the  elements  to  correspond  to  each  of  the 
relations  (i),  (2),  (3). 

In  addition  to  the  relations  already  given,  we  shall  have  need  of 
the  following  equation,  due  to  Euler  : 

(rt  +r3  +  ,)l  -  (r,  +  r3  -  s)*=  3(/s  -  /,) 
f  =  (*3  -  XiY  +  (y3  -y^  +  (z3  -  z^ 

=  pl-  20A  cos  ft  cos  (A,  -  Z,  )  +  2H  +  2PI£3  cos  ft  cos  (A,  -  Z3)  (4) 

+  P3  -    2P3^3  COS  ft  COS  (\»   —  A)  +   2P3#*  COS  ft  COS  (X3  -   Z0 


where  s  is  the  chord  connecting  the  first  and  third  positions  in  the 
orbit.  The  quantities  /3,  t^  are  given  here  in  the  units  used  in  the 
first  part  of  the  paper  ;  and  it  is  understood  that  /3  is  larger  than  /x  ,  so 
that  /3  —  /!  is  positive. 

20.  Plan  of  procedure.  —  From  the  relations  already  set  forth,  it  is 
proposed  to  deduce  formulas  expressing  the  variations,  in/,  O,  /,  o>  and  II 
due  to  a  variation  in  the  observed  co-ordinates  A,,  A,,  A3,  ft,  ft,  and  ft. 
In  this  work  /,,  /2,  /3  are  considered  constant,  as  are  also  the  quantities 
RI  ,  R^  ,  R^  ,  Zt  ,  Z2  ,  Z3  ,  which  are  obtained  from  the  Ephemeris  and 
depend  directly  on  /,,  /2,  /3.  It  is  evident  that  rlt  ra,  r3,  pt,  p2,  p3, 
vi>  V2>  V3>  which  occur  in  the  relations,  are  functions  of  A,,  A2,  A3,  ft, 
ft,  ft,  and  will  vary  with  these  co-ordinates.  Hence,  incidentally, 


1 8          CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 

formulae  are  derived  showing  variations  of  these  in  terms  of  the 
observed  co-ordinates.  One  thing  further.  The  quantities  j=-^- 

and  Y~^ — ^  are  functions  of  Xx,  X3,  X3,  ft ,  etc. ;  but  when  these  ratios 

[ftt  r**\ 

are  computed  by  means  of  series,  it  is  easily  seen  that  the  above  quan- 
tities enter  only  in  the  higher  terms  of  the  series — terms  which  are 
small  and  hence  neglected  on  first  approximation.  It  will  remain  to 
show  that  we  may  always  neglect  such  terms  in  taking  the  partial 
differential  coefficients  of  mf,  M' ,  etc.,  in  respect  to  the  observed 
co-ordinates.  This  is  taken  up  later. 

It  resolves  itself  to  this,  then  :  Each  of  the  elements  p,  O,  o>,  /,  and 
II  is  a  function  of  the  six  variables  Xx,  X2,  X3,  ft,  ft,  ft  ;  but  it  is 
found  that  the  work  of  derivation  of  the  formulae  naturally  divides 
itself  into  three  parts.  Thus  we  may  get  expressions  to  determine 
8/,  8O,  8o>,  8/,  and  811  where  these  variations  arise  from  a  variation  8X, 
in  Xj  and  8ft  in  ft  of  the  first  position.  A  second  set  of  formulae  will 
determine  8^>,  8O,  etc.,  where  variations  of  like  character  are  given  to 
the  observed  co-ordinates  of  the  second  position  ;  finally,  a  similar  set 
are  obtained  for  the  third  position.  We  take  up  the  work  in  the  order 
just  indicated. 

21.  The  variations  8XX,  8ft.— The  equations  (i),  (2),  and  (3)  are 
all  of  the  type 


Hence  we  get  for  each  of  these 

,    S  =  v     (5) 


where  8p3,  8p,  are  changes  in  p3,  pt  for  the  variations  8ft  and  8XX  in  the 
arguments  ft  and  Xx.  From  (i)a,  (2)^,  (3)^  we  get  the  following 
expressions  for  the  partial  derivatives  in  (5),  where  we  count  the  ratios 
of  the  triangles  as  independent  of  ft  and  X,  according  to  the  remarks 

of  the  previous  article. 

. 

Q  ii*M 

am  om 


(6) 


BM'  _  [>2,  rj  cos  ft  cos  ft  sin  (X2  -  Z2)  -f  sin  ft  sin  ft  sin  (Xt  —  Z2) 

"9ft"  ~  [>«,  '.]  sin  ft  cos  ft  sin  (X3  -  Z2)  -  sin  ft  cos  ft  sin  (X2  ~  Z2)  ( 7) 

%M'  =  |>2 ,  rj  sin  ft  cos  ft  cos  (X,  -  Z2 

9^i         [^t »  ''J  sin  ft  cos  ft  sin  (X  —  Z2)  —  sin  ft  cos  ft  sin  (X2  —  Z2) 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  19 

=  |>2  ,  r  J       cos  ft  cos  ft  cos  (A2  -  Z2)  +  sin  &  cos  (A2  -  Z2 
~     [r,  ,  rj  sin  ft  cos  ft  cos  (A3  -  Z3)  -  sin  ft  cos  ft  cos  (A,  -  Z2)  ' 
=  [r.,  rj  _  cos  ft  sin  ft  sin  (A,  -Z0)  __  (8) 
[rz  ,  rj  sin  ft  cos  ft  cos  (A3  -  Z3)  -  sin  ft  cos  ft  cos  (A,  -  Z2)  ' 


[>,  ,  rj  cos  ft  sin  (A3  -  A,)   ' 


[rr,  rJcosftsin(AI-A2) 
Using  the  second  relation  of  (4),  we  obtain 


where  we  have 

BlfiiJI' 

|£  =  I  [Pl  -  Xl  cos  A  cos  (X,  -  Z.)  +  ^?3  cos  ft  cos  (A,  -  Z3)]  ; 

°pi         € 

9j       i 

ST  =-|>3-^3cosAcos(A3-  Z3)  +  ^  cos  ft  cos  '(A3-  Z,)]  ; 

Ps       ^  (n) 

8j        i  o" 

g^  ==  -  [ft*;  sin  ft  cos  (A,  -  Zx)  -  Pl^?3  sin  ft  cos  (Ax  -  Z3)]  ; 

a 

gj-  =  7  [Pl^  cos  ft  sin  (A,  -  Z.)  -  PtX3  cos  ft  sin  (A,  -  Z3>]  ; 

By  means  of  (5)  we  may  eliminate  8p3  from  (10),  and  get 


-f-  M 


From   the  first  relation  of  (4)  we  get,  where  for  brevity  we   put 

r,  +  r  =  K  , 


^  (   } 

VK+S     VK-SV 

This  equation  may  be  written 

'  _ 

r   .     „     I          /S\ 

^r  +  ^i-y 
8^=  ~T~ 


20          CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 

Kow,  where  z>3—  vt  <  180°,  which  includes  all   practical  cases  in 
which  it  is  possible  to  make  use  of  the  expansions  treated  of  in  Part  I, 

—  will  always  be  less  than  unity;  and  hence  we  may  write 
AT 


3-5 


2*  K 

From  this  it  follows  that  (13)  may  be  written 

2K          Ti  s        i       i/J\3  •  * 

8  A  =  ---  dJ  -f  I  -  —  -f-  -  :    ~  I  ~F  I  ~r  '   '   '~ 
j-  L2  ^       2       4  VAT/  « 


The  series  on  the  right  will,  in  all  legitimate  cases,  be  very  convergent, 
owing  to  the  fact  that  there  must  always  be  the  strong  inequality  s  <  K. 
In  general  not  more  than  three  terms  will  be  necessary  (usually  two 
are  sufficient)  where  six-place  logarithms  are  used. 

By  use  of  (12  and  14)  we  are  able  to  express  8  (rt  -f-  r3)  directly  in 
terms  of  variations  of  ft,  Xlf  and  px.  We  do  not  write  out  the  result, 
which  is  very  simple.  It  may  be  noticed  here  that  in  (14)  a  singularity 
enters  into  the  coefficient  of  8s  when  s  approaches  zero.  In  this  case 
the  value  of  8J£  would  depend  on  the  first  term  on  the  right  and  would 
be  large.  The  same  difficulty  is  found  in  the  value  of  8s  itself,  since, 
as  seen  from  (n),  the  partial  derivatives  in  respect  to  plf  p3,  Xr,  ft  each 
become  large.  From  this  we  conclude  that  when  the  observations  are 
taken  at  very  short  intervals  in  the  orbit,  then  the  computed  value  of 
r,  -f-  r3  will  be  very  inaccurate,  owing  to  the  errors  of  the  observations. 

From  a  well-known  trigonometrical  relation  upon  the  triangle 
whose  sides  are  p,  r,  and  R  we  have  the  two  relations  for  the  first  and 
third  positions  of  the  comet  and  earth  in  respect  to  the  sun  : 

r?  =  p*  -  2Pl£,  cos  ft  cos  (X,  -  L,}  +  /?,-  ,  ,     . 

r3*  =  p3*  -  2p3tf3  cos  ft  cos  (X3  -  Z3)  +  R*  . 

From  (15)  we  obtain 

8rt  =  -5-'  Spx  +  —^  8ft  -f-  £— *  8\,  , 

op  ap,  OA,  ,   ,s 

*.-£*• 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  2  I 

where  we  have 


^  =  -  [Pl  -  R,  cos  ft  cos  (Xr  -  A)]  , 
d/ox        rx 

Jj  =  ^  [P,  -  *s  ^s  ft  cos  (X3  -  /,)]  . 
By  means  of  (5)  equations  (16)  may  be  written 

fr.  =  r'fc  +!£«&+£».  . 
dp,  dft  dA 


- 
I+P'        I+ 


Now  from  (14),  when  combined,  as  suggested  above,  with  (12),  we 
have 

Sr^SK-Sr*  ,  /     x 

8^  =  8^-^  , 

where  §K  involves  only  the  differentials  Sft,  8Xt,  8/ot  in  linear  expres- 
sions. Hence,  by  combining  (18)  and  (19)  we  may  get  8r3  and  8rx  in 
expressions  of  the  form  : 

'1  ,  (     . 

,  , 

where  <:,  c'  ,  d,  d'  are  constants.  By  substituting  these  back  into  (18) 
we  may  also  get  8^  in  the  form 

8p3  =  .8ft  +  ^8XI  ,  (21) 

where  e  and  e'  are  constants  not  involving  8ft  or  8XX.  If  we  place 
this  last  in  (5),  we  also  get  8p3  in  the  form 

XI  ,  (22) 


where  /,  /'   do  not  involve  8ft  or   8X,.     These  auxiliary  relations, 
8r,,  8r3,  8/5t,  8p3,  are  very  useful  for  the  work  which  follows  in  getting 
the  variations  of  the  elements/,  O,  o>,  /',  and  II  in  terms  of  8ft,  8XX. 
We  start  with  the  equations  (7)  of  Part  I,  as  follows  : 


22          CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 
p  =  r  (  i  -f-  cos  v)  =  2r  cos2  -  , 


(23) 


^ 

tan  -  4-  -  tan3  -  =  2 


. 
2  /I  . 

From  the  first  of  these  we  get  for  the  first  and  third  positions 


where  the  radicals  are  to  be  taken  positive,  if  -  is  the  first  or  third 

quadrant;  and  negative  if  in  the  second  or  fourth  quadrant.     By  use 
of  these  relations  we  obtain  from  the  second  relation  of  (23) 


Eliminating  n  from  (24)  and  clearing  of  fractions,  we  get 


2/3—  /  -  V  2r-p  -f  i/(2rs-/)3  -  V(2rt  -/)3 

^^s-O]  •      (25) 

By  giving  variations  Srj  and  8/*3  to  ^  and  r3  respectively,  we  obtain, 
after  collecting  and  simplifying  and  solving  for  S/, 


-/-('i  -/)•»!-> 

=  Sr3  .          (26) 


In  (26)  it  must  be  kept  in  mind  that  the  radicals  are  positive  or  nega- 
tive according  as  tan  -  is  positive  or  negative.     This  equation,  when 

the  auxiliary  equations  (20)  are  used,  gives  8/  as  a  linear  function  of 

8&,  &,. 

From  the  relations 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  23 


we  derive  by  differentiation 

*T-/ 


,  sn 


=  -  8r3  —  r,  sin  v38v3  , 


from  which 


o_. .x A.** 

1      r*  sin  vs     T       r,  sin  z>,  ' 

•       ^^,-i-C-         (27) 

r3  sin  z>3  r3  sin  z>3 

By  means  of  (27),  (26),  and  (20),  we  may  get  8^,  8z»3  expressed  as  linear 
functions  of  8ft,  8\T. 

If  we  denote  the  argument  of  latitude  by  u .=  ^-{-w,  we  have  then 
the  relation 

from  which 

From  (6)  of  Part  I  and  corresponding  relations  expressing  the  car- 
tesian co-ordinates  in  terms  of  geocentric  longitude  and  latitudes, 
we  get 

z 
8^  =  —  8rx  -f-  rz  sin  ut  cos  io/  +  r^  cos  u1  sin  tout  ,  /     \ 


—  sin  pdpz  -f-  p,  cos  ^d/J,  , 

z 

=  —  8r.  +  r,  sin  w,  cos  101  4-  f,  cos  w_  sin  tou,  ,  x     x 

r3  (30) 


By  means  of  (28),  (29),  and  (30),  we  may  get  8/,  8wx,  8«3,  in  linear 
functions  of  8/?x  ,  8X,  . 
We  also  obtain 

x 

«£==—  Brt  —  ^SO-f-z,  sin  O8/—  /•,  (sin  «x  sin  O  —  cos  wx  cos  O  cos  i)^,  , 


Sxt=  cos  ft  cos  Mp,  —  px  sin  ft  cos  Xt8ft  —  pl  cos  ft  sin  X18AI  . 

By  means  of  these  equations,  or  the  corresponding  ones  for  the  third 
position  which  may  be  used  as  a  check,  we  may  get  8O. 
From  the  relation  o>  =  u  —  v,  we  obtain 


24          CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 

Finally  from 

v.i         tv       2(/  —  II) 
tan--f  -tan^-  =  -^  -  '    , 

2^32  # 

we  obtain 

m=   -^(A-n^-^sec^'Sv,, 


42 

With  these  we  have  a  complete  set  of  formulae  by  which  the  variation 
of  any  one  of  the  five  elements  of  the  cometary  orbit  is  expressible  as 
a  linear  function  of  Sft  and  8XZ  . 

22.  The  variations  8ft  8X2.  —  If  now  errors  are  made  in  the  observa- 
tion of  the  second  place,  these  errors  will  also  have  an  effect  upon  the 
elements.  We  consider  the  errors  thus  caused  and  give  formulae  for 
their  computation.  In  these  formulae  it  is  to  be  noticed  that  we  use 
the  expressions  8/Oj,  8p3,  Sr1}  etc.,  to  designate  the  variation  of  the 
quantities  pl}  p3,  rzt  etc.,  only  so  far  as  ft  and  X2  are  concerned.  They 
must  not  be  confused  with  the  same  expressions  used  heretofore,  where 
only  Xx  and  ft  were  considered  to  vary.  The  apparent  ambiguity  is 
justified  by  the  simplicity  which  this  usage  gives  to  the  writing  of  the 
two  sets  of  formulae.  Furthermore,  we  content  ourselves  here  by 
simply  writing  down  the  results  of  the  derivations.  This  is  done 
because  the  actual  work  of  obtaining  the  equations  is  very  similar  to 
that  pursued  in  the  last  article,  and  where  any  divergence  occurs  the 
formulae  themselves  enable  one  to  see  the  method  used.  Attention  is 
again  called  to  the  approximation  used  in  the  values  of  the  ratios  of 
the  triangles  in  obtaining  the  partial  derivatives  of  M  and  m  in  respect 
to  ft  and  X2. 

To  start  with  we  have  the  equation  analogous  to  (5)  : 


x          M*      •         */?    ixi  -r> 

8/>3  =  M8Pl  +  _  8ft  +  g^-  8X2  +*  jg-  Sft  +  /».§£-  8A2  ,       (34) 

where  the  partial  derivatives  are  gotten  from  the  expressions  for  m'  , 
mn  ',  m"',  M'  ,M"  ,  M'"  given  in  (i)  a,  (2)  a,  and  (3)  a;  owing  to 
their  length  we  omit  them  here.  It  is  to  be  noticed,  however,  that 

BM'"  . 
here  —f^  —  is  zero. 


=  -  [P  -  jR,  cos  ft  cos  (X,  -  Zz)  +  R3  cos  ft  cos  (X,  -  Z3)]  8Pl 

(35) 
+  -  [p3  —  7?3  cos  ft  cos  (X3  -  L3)  +  ^  cos  ft  cos  (X3  -  Zf)]  8p3  . 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  25 

2K  i  I    3.5---(2"-3)/ 


s 

(36) 

where  K—r^  -\-  r3. 

8rt  =  — -  [px  —  RI  cos  pl  cos  (Xz  —  Z,)]  , 

(37) 


*3  ==  ~^  [Ps  ~~  -^3  COS  A  COS  (^3  

:r!5~t3)  (38) 


(r3  -/)  V  2r.  -/  -  (r,  -/)  I/  2r3  -/ 

8rx  — ^3      2— — ^ 8r,  .      (39) 

(r,  -  p)V,rt-p-(r,-p)  V  *rt  -p     ' 


*  sin  vx  rt  sin  vz 

(40) 


8^  =  Sv3  —  8vt  .  (41) 

—  8rT  +  r,  sin  2/t  cos  /8i  +  ^  cos  a,  sin  t'8ut  ,  .     . 

ri  (42) 


Bz3=  —  8r3  -f-  r3  sin  «3  cos  i'S/  -(-  ^3  cos  «3  sin  /8w3  , 

rs  (43) 


X 

8xt  =  —  8rt  —  y^l-\-zl  sin  H8/  —  rx  [sin  ut  sin  li—  cos  w,  cos  Q  cos  /]  8wt  , 

^x  /     "k 


26          CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 

This  completes  the  list  of  formulae  for  obtaining  the  variations  of 
the  elements  in  terms  of  the  variations  8ft,  8X2. 

23.  The  variations  8ft,  8X3.  —  It  is  easily  seen  that  the  formulas  in 
the  case  of  errors  in  the  third  position  will  be  very  similar  to  those  of 
the  first  position.      This  is  due  to  the  fact  that  in  the  computation  of 
the  elements  the  co-ordinates  of  the  first  and  last  positions  play  very 
similar  roles.     We  do  not  give  the  formulas  for  the  last  position  here, 
since  it  would  unnecessarily  prolong  this  discussion. 

24.  Dependence  of  the  ratios  of  the  triangles  on  XIt  ft,  X3,  ft.  —  We 
come  now  to  a  question  left  over  from  a  previous  article.     It  is  as  to 
the  dependence  of  the  ratios  of  the  triangles  upon  the  co-ordinates  of 
the  first  and  third  positions.     The  question  reduces  to  one  as  to  the 

dependence  of  r2  and  —  2  upon  the  above  co-ordinates.     That  this  is 

true  is  foreshadowed  by  the  terms  of  the  series  written  out  on  the  right 
of  (24),  Part  I  ;  but  it  is  also  capable  of  derivation  from  the  Newtonian 
law  of  motion  itself  that  such  is  the  case.  For  we  have  in  general 


and  by  means  of  this  relation  all  the  higher  derivations  of  r2  in  the 
development  of  the  ratios  are  reducible  back  upon  r2  and  —^  . 

Now,  the  quantities  r2  and  —^  are  by  the  manner  of  their  computa- 

tion in  Olbers's  method  each  functions  of  ft,  X,,  ft,  X3  as  well  as  of  X2, 
or  ft,  or  both  of  the  latter  two,  according  to  the  particular  one  of  the 
equations  (i),  (2),  or  (3)  which  have  been  employed.  But,  in  taking 
the  partial  derivations  of  M  and  m  in  respect  to  the  co-ordinates  Xt, 
ft,  X3,  etc.,  we  have  assumed  the  ratios  to  be  independent  of  these 
quantities.  It  is  necessary,  then,  to  justify  this  assumption. 

To  begin  with  it  can  be  justified  only  from  the  standpoint  of  its  being 
a  near  approximation  to  the  truth  when  the  computation  has  been 
carried  out  in  the  method  described  in  article  17,  namely,  when  the 
time  intervals  have  been  so  chosen  as  to  give  series  (24),  Part  I,  the 
proper  convergency.  As  we  have  already  remarked,  this  convergency 
should  be  so  rapid  that  for  any  given  case  the  remainders  after  the 
second  term  will  be  of  the  order  of  smallness  of  the  lowest  decimal 
place  which  is  omitted  in  the  process  of  computation.  In  order  to 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  27 

verify  the  statement  made  above  as  to  the  validity  of  the  approxima- 
tion in  question,  we  prove  the  following  theorem,  which  we  then 
proceed  to  apply. 

Theorem:  The  variation  8r2  arising  from  uncertainties  in  A,,  A2,  A3, 
Pi,  &,  ft  is  at  least  of  the  order  of  smallness  of  8^,  and  8Q. 

For,  in  the  same  manner  as  we  obtained  (33),  we  get,  after  reduction, 

811-.          (48) 


From  the  relation  r2  (i  -f  cos  v2)=#,  we  get 


sn 


whence,  by  use  of  (48), 

*>.  =J  ¥  +  ^l  [<n  -  O  sin  ..  -  5^]  SH  .  (49) 

An  inspection  of  the  right  member  of  (49)  at  once  makes  the  truth  of 
the  theorem  apparent. 

Now,  in  getting  the  partial  derivatives  in  (7),  (8),  (9),  terrrs  are 
omitted,  among  others,  of  the  following  forms  : 


.  ,. 

where  clt  cz  are  quantities  easily  determined  from  equations  (i),  (2), 
(3),  and  which  are  small  when  the  problem  is  taken  under  proper  con- 
ditions. Now,  for  any  particular  and  approximate  set  of  quantities 
(V3  —  /2),  (/2  —  /,),  and/,  it  is  known  that  the  coefficients  of  (50),  aside 
from  the  factors  £,,  ^2,  are  small  —  not  amounting  to  more  than  a 
digit  in  the  third  or  fourth  decimal  place.  For  the  discussion  of  this 
in  a  particular  case,  see  Moultorfs  Introduction  to  Celestial  Mechanics, 
art.  208.  If  we  accept  these  statements,  which  are  shown  to  be  true  by 
mere  mechanical  computation,  it  results  that  the  terms  omitted  in  art. 
21  are  not  in  general  of  appreciable  size.  In  any  case  where  the 
formulae  of  this  paper  are  used  for  computation,  the  value  of  these 
terms  may  be  computed  after  the  manner  of  Dr.  Moulton's  discussion 
just  cited,  and  their  probable  value  obtained.  Should  they  be  of  con- 
siderable size,  they  must  be  taken  into  account  in  the  computation  of 
the  uncertainties  of  the  elements. 


28 


CONVERGENCE  OF  SERIES  USED  IN  DETERMINATION 


25.  Computation  of  the  errors. — An  example:  The  following  applica- 
tion of  the  formulae  (i)  to  (33)  is  here  given  in  abstract  as  an  example 
to  accompany  the  theory  here  set  forth.  The  computation  of  the 
elements  was  made  by  Dr.  K.  Laves  and  members  of  his  class  in  the 
summer  of  1900.  The  remaining  computation  was  done  by  myself. 
Most  of  the  results  were  checked  by  various  devices  invented  for  the 
occasion,  some  of  which  are  mentioned  in  the  previous  discussions. 

The  orbit  was  that  of  the  comet  Borelly-Brooks  (1900.6).  The 
observations  were  made  at  the  Chamberlain  Observatory  of  the  Uni- 
versity of  Denver  by. Mr.  Ling  with  the  twenty-inch  refractor.  They 
were  as  follows  : 

UNIVERSITY  PARK  M.T. 
X  =  6h  59m  47?63  .0  =  39°  4<>'  36-4 


Comp.* 

No.  Comp. 

aO^-o* 

Greenwich  M.  T. 

1900,  July  24d  I3h37m2is 
August  2d  1  2h    9m56s 
August  6d  i  oh  26m  1  6s 

Lamb  1541 
Bonn  2544 
Lamb  («3)  1  385 

20  ;  6 
20;  8 
20  ;  6 

16'   47^2 

2'    28  ?4 
2'    27?9 

2oh  37™  8*63 
I9h  9m  43?63 
I7h  26m  3?63 

Elements  computed  by  class  and  Dr.  Laves  : 

log  q  —  0.006363 
0=327°  59'  59' 


29'  32 

24'    47 

-  II  =  August  3,  1900,  Greenwich  M.  T. 

K 


=I2°    24'    47" 


The  following  values  were  obtained  for  co-ordinates  by  Dr.  Laves, 
which  I  have  corrected  for  the  notation  used  in  this  paper  and  used 
throughout  in  computation  : 


A,  =  49°  13'  22-6,     &=i3°  28' 


31 


A2  =  54°  20'  46  fo  ,  /£>  =  : 

^3  —  6o°     4'  43fo  ,  $,  =  35-     i     56", 

log  M'  —9.950021  —  10  ,  log  (rt  +  r3)  =  0.308352  , 

log     ^  =  0.007514,  log  ^  =  9. 278413  —  10. 
log      r3  =  o.< 


We  have  numbered  the  equations  giving  the  results  for  each  step  so 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  29 

as  to  correspond  with  the  formulae  from  which  they  are  derived  in  art. 
21.  Where  the  coefficients  are  logarithms,  we  have  indicated  it  by 
placing  the  left-hand  member  in  parenthesis,  thus  (8r,),  (8r3),  etc. 

8/33=1.05854  8/0,      -2.266778)8,    -O.I470438A,  .  (5') 

8^  =  3.496908/3,     —3.779478/3,    +  0.20662  8AX.  ,. 

(Bs)  =  0.543683 8p,  —0.577421  8ft +  9.315172  SA,  . 

8AT=  —  74.76678  8pz  + 80.80672  8ft  — 4.41772  8At  .         (14') 

8r,  =  0.236288  8p,  -0.577421  8ft  +  0.032823  SA,  , 
Sr3  =  0.2106228  8p,+  0.022165  8/8,  —  0.416143  8X, . 

8r3  =  —  74.97740  8px+ 80.78458/3,    -4.00I588A,  ,  , 

8r,=—  75. 00307  8px  +81.312728)8,  —  4.38489  8X,  . 

From  (i8r)  and  (19')  we  get  one  and  the  same  result,  as  follows  : 

8pT  =  108080  Sj8r  — 0.052766  8X,  .  (5") 

Putting  this  in  (5') 

8/o3  =  —  1.12271  8)8,  —  0.202898  SA,  . 

Putting  these  in  (18')  and  (19'), 

8r,  =  0.24981  8)8,  — 0.42726  8A,  ,  .     g. 

8r3  =  —  0.25062  8ft  — 0.04530  SA,  . 

The  uncertainty  in/  was  found  to  be 

8p  =  —  0.058051  8ft  — 0.437117  8A,  .  (26') 

The  following  then  were  obtained  in  the  order  given  : 

8^,  =  -5-31986  8ft  +  3.96961  8A,  , 
8?3  =  -5.i87358ft  +  4.o6573  8AX  . 

8#3  — 8^  =  0.13251  8ft  +  0.09612  8A,  .  (28') 

8/=  —  15.0415  8ft  —  2. 4690  8A,  .  (3°') 

8»f=  1.6237  8ft +  0.18104  8A,  , 
8^=1.75628 8ft  +  0.2772  8AX  . 

m  =  0.6799  8ft  —  0.0763  8AX  .  (31 ;) 

i8n  =  -22i.   8ft+i7o.  8A,  .  (33r) 

^n  (33')  tne  dig^  in  units  place  is  uncertain  by  3  in  each  coefficient. 
We  finally  get  for  the  uncertainty  in  the  longitude  of  the  perihelion 

8cu  =6.9436  8ft  —  3.7886  8A,  .  (32') 


30          CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 

In  order  to  get  the  amount  of  an  error  of  an  element  for  a  given 
error  in  an  observed  co-ordinate,  we  need  only  substitute  the  values  of 
8f}t  and  &Xt  which  are  allowable  from  the  nature  of  the  conditions 
under  which  the  settings  of  the  instrument  were  made.  Suppose,  for 
instance,  that  8/3t  equals  to  one  second  of  arc.  This,  in  circular 
measure,  will  correspond  to  0.00000485.  Then  by  (33')  an  error  of 
one  unit  in  the  third  decimal  place  would  result;  while  by  (27 ')  the 
error  would  be  confined  to  the  seventh  decimal  place,  and  in  this 
latter  case  be  perfectly  harmless  for  ordinary  six-place  tables. 

As  to  the  amount  of  error  actually  probable,  various  observers 
differ  in  their  estimates.  LeVerrier  says:1  "Experience  has  proved 
that,  for  stars  with  feeble  light,  errors  of  4"  or  5'  do  not  exceed  the 
limits  of  the  possible  nor  indeed  of  the  probable."  In  this  he  was 
referring  to  the  observation  of  the  asteroids.  Dr.  Hussey,  of  the  Lick 
Observatory,  considers  this  estimate  far  too  large  for  the  case  where  a 
star  and  an  asteriod  are  being  compared  with  large  modern  instru- 
ments. However  he  states  that  "for  comets  the  conditions  vary 
greatly;  and  for  those  without  visible  nuclei,  large  and  cloudlike ;  in 
such  cases,  even  under  favorable  atmospheric  and  instrumental  condi- 
tions, "  one  might  be  doing  very  well  indeed  if  he  kept  his  errors  of 
observation  below  10."  Dr.  Barnard,  of  Yerkes  Observatory,  esti- 
mates, with  Dr.  Hussey,  that  ofi  ought  to  be  the  limit  of  error  for  an 
asteroid  of  the  fainter  kind  if  the  conditions  are  favorable;  but  he 
says:  "A  comet's  position  is  far  more  uncertain;  and  discordances  of 
several  seconds  of  arc  are  not  unusual  in  the  work  of  good  observers. 
Much  depends,  of  course,  on  the  presence  or  absence  of  a  nucleus 
and  the  faintness  of  the  comet ;  but  in  general  comet  observations  are 
distressfully  discordant" 

The  above  statements  of  Dr.  Hussey  and  Dr.  Barnard  are  from 
personal  letters  to  the  writer  in  answer  to  inquiries  in  regard  to  the 
accuracy  of  comet  observations  with  the  best  modern  instruments.  If 
we  admit,  then,  that  the  errors  in  observation  may  vary  from  one  to 
ten  seconds  of  arc  in  the  case  of  comets,  the  results  (5')  to  (32 ')  show 
that  the  computed  elements  will  be  discordant,  owing  to  such  errors 
in  a  single  co-ordinate,  as  follows : 

Element/  is  discordant  in  sixth  or  seventh  decimal  place, 
Element  *  is  discordant  in  fourth  or  fifth  decimal  place, 
Element  O  is  discordant  in  fifth  or  sixth  decimal  place, 
Element  II  is  discordant  in  third  or  fourth  decimal  place, 
Element  <o  is  discordant  in  fourth  or  fifth  decimal  place. 

'  C./?.,  Vol.  XXV,  p.  573. 


OF  ELEMENTS  OF  PARABOLIC  ORBITS  31 

These  results  might  be  very  much  decreased  when  combined  with 
like  errors  in  the  measurements  of  the  other  five  observed  co-ordinates; 
or  they  might  so  balance  each  other  that  very  little  discordance  would 
be  present  in  the  computed  elements.  As  to  this,  however,  we  can 
only  say  that  they  are  still  uncertain  by  the  amount  represented  by 
the  combined  effect  of  all  the  errors  arising  from  all  of  the  observed 
elements. 


BIOGRAPHICAL. 

WILLIAM  ALBERT  HAMILTON  was  born  May  9,  1869,  near  Zanes- 
ville,  Ind.,  and  received  his  elementary  education  in  the  district 
schools,  which  he  attended  during  the  winter  months.  He  was  pre- 
pared for  college  at  Roanoke  Classical  Seminary  of  Roanoke,  Ind., 
and  in  the  preparatory  department  of  North  Manchester  College,  of 
North  Manchester,  Ind.,  and  graduated  from  the  collegiate  department 
of  the  latter  school  in  1892.  In  1894  he  entered  Indiana  University 
and  graduated  from  the  Department  of  Liberal  Arts  in  1896  with  the 
degree  of  A.B.  In  this  course  Mathematics  was  his  major  subject. 
During  the  year  1898-99  he  studied  in  both  the  University  of  Chicago 
and  Indiana  University,  receiving  the  degree  of  Master  of  Arts  from 
the  latter  institution  at  the  close  of  the  collegiate  year,  with  Mechanics 
and  Astronomy  as  major  subject.  His  thesis  was  upon  "The  Path  of  a 
Particle  Which  is  Subject  to  a  Central  Force  Which  Attracts  Inversely 
as  the  Fourth  Power  of  the  Distance." 

Mr.  Hamilton,  during  the  intermission  of  his  collegiate  and  pre- 
paratory studies,  taught  in  public  school  and  college  work.  The 
positions  held  up  to  and  including  the  year  1899-1900  are  as  follows  : 

In  1889-90,  Teacher  in  Common  School;  1891-92,  Instructor  in 
Latin  and  Algebra  in  North  Manchester  College;  1892-94,  Principal 
of  High  School  at  Butler,  Ind.;  1896-98,  Superintendent  of  Schools 
at  Hebron,  Ind.  ;  1899-1900,  Teacher  of  Mathematics  in  the  California 
School  of  Mechanical  Arts,  San  Francisco,  Calif. 

In  1900  Mr.  Hamilton  re-entered  the  University  of  Chicago,  taking 
lectures  in  Astronomy  and  Mathematics.  Including  the  time  pre- 
viously spent  in  this  institution,  he  took  courses  as  follows :  under  Dr. 
Laves  :  Theory  of  Orbits,  Analytic  Mechanics,  Special  Perturbations, 
Absolute  Perturbations,  Theory  of  Attractions  of  Heavenly  Bodies, 
Spherical  and  Practical  Astronomy;  under  Dr.  Moulton  :  Method  of 
Least  Squares,  Physical  Astronomy,  Problem  of  Three  Bodies,  Intro- 


32          CONVERGENCY  OF  SERIES  USED  IN  DETERMINATION 

duction  to  Celestial  Mechanics,  Lunar  Theory;  under  Mr.  Lunn  :  The 
Problem  of  n  Bodies ;  under  Dr.  Boyd  :  Theoretical  Mechanics. 

In  Mathematics  he  took  lectures  as  follows  :  under  Professor 
Moore  :  Projective  Geometry  ;  under  Professor  Bolza  :  Definite  Inte- 
grals, Theory  of  Functions  of  a  Complex  Variable  and  Elliptic 
Functions  ;  under  Professor  Maschke  :  Fourier's  Series  and  Elliptic 
Integrals,  Theory  of  Functions  of  a  Complex  Variable  and  the  Theory 
of  Invariants ;  under  Dr.  Slaught :  Definite  Integrals  and  Differential 
Equations. 


OF  THE 

I    UNIVERSITY 

I  V  OF 


